Subgroup ($H$) information
| Description: | $C_2^2.\OD_{32}$ |
| Order: | \(128\)\(\medspace = 2^{7} \) |
| Index: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| Generators: |
$\left(\begin{array}{rr}
27 & 26 \\
6 & 21
\end{array}\right), \left(\begin{array}{rr}
1 & 16 \\
8 & 9
\end{array}\right)$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_4^4.C_{24}$ |
| Order: | \(6144\)\(\medspace = 2^{11} \cdot 3 \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and metabelian (hence solvable). Whether it is monomial has not been computed.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_4^2.(C_2^4\times C_{12}).C_2^6.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2^7.D_4$, of order \(1024\)\(\medspace = 2^{10} \) |
| $W$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $96$ |
| Number of conjugacy classes in this autjugacy class | $16$ |
| Möbius function | not computed |
| Projective image | $C_2^5.A_4$ |